What is the differences between Object AXES and Edit Mode AXES
In object mode you can read the orientation of the whole Armature object (which is the orientation of its origin, and can also be in a location in which there is no bone), in edit mode you can read every single orientation of every single bone of the armature.

The orientation of every bone is displayed at the tip of the bone, but I think is more intuitive to think at it located at the root of the bone, where the rotations will happen.